[schindlabua]

I like thinking of this concept via free constructions.

As is somewhat commonly known the free Monoid is the List type; monoids are not commutative so we get a sense of "direction", like a list has a start and an end.

If we add commutativity and look at free groups, we find they are equivalent to multisets.

If we take associativity away from monoids and look at free semigroups, we get binary finger trees, I think?

In some sense removing constraints from the binary operator results in more general free types. Would be interesting to find what free construction makes digraphs but I have to bounce.